Abstract

In the conformal field theory (CFT) approach to the quantum Hall effect, the multi-electron wave functions are expressed as correlation functions in certain rational CFTs. While this approach has led to a well-understood description of the fractionally charged quasihole excitations, the quasielectrons have turned out to be much harder to handle. In particular, forming quasielectron states requires non-local operators, in sharp contrast to quasiholes that can be created by local chiral vertex operators. In both cases, the operators are strongly constrained by general requirements of symmetry, braiding and fusion. Here we construct a quasielectron operator satisfying these demands and show that it reproduces known good quasiparticle wave functions, as well as predicts new ones. In particular we propose explicit wave functions for quasielectron excitations of the Moore-Read Pfaffian state.
Further, this operator allows us to explicitly express the composite fermion wave functions in the positive Jain series in hierarchical form, thus settling a longtime controversy. We also critically discuss the status of the fractional statistics of quasiparticles in the Abelian hierarchical quantum Hall states, and argue that our construction of localized quasielectron states sheds new light on their statistics.
At the technical level we introduce a generalized normal ordering, that allows us to ”fuse” an electron operator with the inverse of an hole operator, and also an alternative approach to the background charge needed to neutralize CFT correlators. As a result we get a fully holomorphic CFT representation of a large set of quantum Hall wave functions.

pacs:

Ever since Laughlin presented his famous wave function describing the ν=1/m quantum Hall (QH) states (1), the construction of trial many-body wave functions has been an important tool to study the properties of the many exotic, strongly correlated states in the QH system(2); (3). In a series of recent papers, it was shown how to use CFT to construct a large class of wave functions describing a subset of the hierarchical QH ground states as well as their quasiparticle excitations(4); (5). Included in this class are the Laughlin wave functions, the wave functions in the positive Jain series ν=2n/(2pn+1), and candidate wave functions for the observed states at ν=4/11 and ν=6/17(6).

The general idea of the CFT construction of QH wave functions is to represent the particles (electrons and quasiholes) of the theory by holomorphic vertex operators in a rational CFT, V∼ei∑jcjφj where the φj’s are compact, massless scalar fields. Candidate many-body wave functions of the corresponding QH state are then expressed as correlators, or conformal blocks, of these operators. Depending on the state in question, the wave functions either take the form of a single correlator, or as an antisymmetrized sum over several conformal blocks. The Laughlin states and Moore-Read Pfaffian(2) are examples of the former, while general states in the Abelian hierarchy fall into the latter class for the following reason: our construction(5) for the
level n hierarchy states involves n different representations of the electron, Vα(z),α=1,...n, involving n compact scalar fields φα(z), with z denoting the position of the particle (we use complex coordinates z=x+iy, etc.);
in the simplest case, that of the Jain series, Vn can be thought of as a describing a particle in the nth composite fermion Landau level. The sum over correlators was thus necessary to antisymmetrize the wave functions among the different representations Vα. Correspondingly, there are n independent quasihole operators Hα(η), describing quasiholes at positions η. Candidate wave functions for the ground states at all levels of the hierarchy were thus expressed as linear combinations of conformal blocks containing electron operators Vα only; quasihole excitations localized at some points ηi are obtained by inserting ∏iHα(ηi) in the correlators.
In the Laughlin case our expressions coincided with the original proposals by Fubini(7), Moore and Read(2), and Wen(8).

Although experimentally, QH quasiholes and quasielectrons play a very similar and equally important role, the CFT description
of the quasielectron required a rather different approach from that of the holes. The reason for this can be intuitively understood as follows: a quasihole, which corresponds to a charge deficit, can be created at an arbitrary position η just by pushing the incompressible electron liquid away from this point. Mathematically this is achieved by inserting the local operator H(η). The creation of a quasielectron amounts to ”pulling in” the electron liquid towards the quasielectron position, ¯η,1
but this cannot in general be done, since there is a finite probability that there is already an electron at this position. This conflict with the exclusion principle is mathematically manifested in singularities in the electron wave function. More precisely, inserting the inverse of the hole operator H−1(η), which does have the correct charge, results in pole terms ∼1/(zi−η) where zi is an electron position. The way out of this quandary, which was described in Ref. (4), is to create the fractional excess charge corresponding to a quasielectron by shrinking the correlation hole around one of the electrons. This amounts to replacing one of the elecron operators Vn with a modifed electron operator, Pn, which turns out to be very closely related to the new electron operator Vn+1 which appears at the next level in the hierarchy. In this way we could construct quasielectrons in any angular momentum state, and from these it was possible to, by hand, form coherent superpositions describing localized quasielectrons.

This asymmetry in the description of localized quasiholes versus quasielectrons is somewhat unsatisfactory; an obvious question to ask is whether there exists an operator P(¯η) which, when inserted into a correlator of Vα’s, directly creates a quasielectron localized at ¯η. In our previous work, it was not clear whether or not this could be done. Although, for the reasons given above, such an operator could not be strictly local, it could still be ”quasi-local”, i.e. the nonlocality is seen only at the magnetic length scale. To clear this out is important; as explained by Moore and Read in Ref. (2), we would, on general grounds, expect quasielectrons to enter the theory in a way very similar to quasiholes. In particular, we would expect the charge and statistics of the quasielectrons to be coded in the properties of this operator, and this could allow for a better understanding of their braiding phases, and thus their quantum statistics.

In this paper we shall explicitly construct such a quasilocal operator, P(¯η), and show that it has all the general properties expected for a quasielectron operator. We demonstrate that it can be used to explicitly construct many-quasielectron wave functions of both Abelian and non-Abelian states. In particular, we present explicit two- and four quasielectron wave functions for the Moore-Read Pfaffian state.

The construction of the operator P involves two important technical developments. The first concerns the treatment of the background charge needed to neutralize CFT correlators. Our method allows us to directly extract the non-holomorphic Gaussian factors, thus leaving us with purely holomorphic wave functions. The second is related to the precise mathematical meaning of ”shrinking the correlation hole around an electron operator”. As described above, this amounts to forming a new local operator by ”fusing” an electron operator with the inverse of a quasihole. This is achieved by a generalized normal ordering (…)gn, which in the simplest case amounts to the conventional normal ordering(10), but in general differs from it.

There is a long-standing controversy about whether the composite fermion (CF) wave functions proposed by Jain(11) fit into the original hierarchy scheme of Halperin(12) and Haldane(13). At a superficial level, the two approaches are quite different. The CF scheme predicts explicit wave functions which compare well with numerical simulations, and the CF phenomenology has been very successful in explaining a wide range of experimental data. The hierarchy wave functions, which are expressed as rather complicated multiple integrals, are much harder to deal with, and the phenomenology is much less developed. On the other hand, the newly discovered states that do not fall into the Jain sequences, fit naturally into the hierarchy schemes but are much less straightforward to understand in terms of composite fermions. Also, in a recent work, Bergholtz and Karlhede proved that when defined on a thin torus or cylinder, the Jain states are precisely condensates of quasiparticles(14).
Using the formalism developed in this paper, we can explicitly write the full wave functions in the positive Jain series in hierarchical form, i.e. as integrals over multi-quasielectron states with appropriately chosen pseudo wave functions. In our opinion, this shows that the CF states are hierarchical in nature.

The paper is organized as follows. In order to settle notation, and to record some formulae that will be important in the following, we start the next section by summarizing some results from Refs. (4) and (5), on the CFT approach to the hierarchical wave functions. This is followed, in the next subsection, by an outline of a method for handling a homogeneous background charge that will prove to be very useful; some technical details of this discussion are referred to Appendix A. This sets the stage for introducing our construction of a quasi-local quasielectron operator in section III, and in order to minimize technicalities we first do this in the simplest case of a single quasielectron in a Laughlin state.
Section IV begins with a general discussion of the status of fractional statistics for quasiparticles in the hierarchical quantum Hall States. Then, specializing again to the Laughlin states, we point out the difficulties encountered when naively extending the methods of the previous section to the case of many quasielectrons, and explain how to proceed. An explicit example, that of the two-quasielectron wave function, is worked out in some detail in Appendix B.
In section V, which is of a more technical nature and not necessary for understanding the rest of the paper, we introduce the generalized normal ordering mentioned above, and give a definition of the operator P which is appropriate for the more general class of hierarchy states constructed in Ref. (4). The explicit hierarchical form of the Jain wave functions is derived in section VI. Finally, section VII discusses the Moore-Read Pfaffian state; here, it is shown how to write down quasilocal quasielectron operators in direct analogy to the Abelian case, and how this leads to explicit two- and four-quasielectron wave functions. We end the paper with some conclusions and a discussion of future directions. A short summary of some of the results of this paper was recently given in Ref. (9).

In order to make this paper reasonably self-contained, we start by summarizing the formalism used in the
CFT description of QH wave functions, as well as recording some formulas that will be used
later. In the second subsection, we discuss an important technical issue, namely how to introduce a background charge in the CFT correlators in a consistent manner.

ii.1 Operators, currents and correlators

As mentioned in the introduction, it has been shown that a large class of quantum Hall wave functions may be expressed as correlators, or conformal blocks, in certain CFTs. In order to review the basic formalism, we start with the simplest case, namely the Laughlin states at ν=1/m=1/(2p+1). In this case there are two normal-ordered vertex operators representing the electron and quasihole, respectively,

V1(z)

=

:ei√mφ1(z):

(1)

H(η)

=

:ei√mφ1(η):,

(2)

and as we shall see in a moment, the wave functions for the ground state and quasihole states can be written as correlation functions of these.
(The normal ordering symbol :: will be suppressed in the following.
For simplicity we will also drop the subscripts on fields and vertex operators for the Laughlin states as long as there are no ambiguities.)
Here, φ is (the holomorphic part of) a free massless boson field, compactified on a radius R=√m and defined by the Euclidean action

S[φ]=∫d2xL=18π∫d2x∂μφ∂μφ

(3)

with d2x=dxdy=12d2z.
The action is normalized to imply the (holomorphic) two point function
⟨φ(z)φ(w)⟩=−ln(z−w).
The vertex operators
satisfy the operator product expansion (OPE)

eiaφ(z)eibφ(w)

=

(z−w)abei(a+b)φ(w)

(4)

+

aa+b(z−w)ab+1∂wei(a+b)φ(w)+O(z−w)ab+2.

Here and in the following, ∂ denotes a holomorphic derivative, while ¯∂ is used for anti-holomorphic derivatives.
Using standard methods(10),
one can explicitly calculate the relevant correlation functions. In particular, the ground state Laughlin wave functions at ν=1/m are given by

Ψ1/m({zi})=⟨N∏i=1V(zi)Obg⟩=∏i<j(zi−zj)me−∑i|zi|24ℓ2,

(5)

where Obg is a neutralizing background charge to be discussed in more detail below. Similarly, quasihole states are constructed by insertions of the hole operator H(η).

The (holomorphic) U(1) charge density current, J(z)=i√m∂φ(z), and the corresponding charge operator, Q=1√m12π∮dz∂φ(z), are normalized so that the electron operators (1) have unit charge, as can be seen from the commutator [Q,V(z)]=V(z); similarly, the charge of the hole operator is 1/m. This U(1) charge is related to the actual electric charge of electrons and quasiparticles, as described in detail in Ref. (4).
Translational invariance of the action (3) implies that only charge neutral correlators are non-vanishing, which is why the neutralizing background charge mentioned above is needed. However, there are additional contraints on the correlators, in particular, a correlator involving the divergence of the current has to satisfy the Ward identity (10):

¯∂ω⟨J(ω)Φ1(z1)...Φn(zn)⟩

=

∑iqiδ2(ω−zi)⟨Φ1(z1)...Φn(zn)⟩,

where the Φi’s are arbitrary fields with charges qi.
Thus, we find that ¯∂ωJ(ω) has support only at the points where a charged field is inserted.
We shall make use of this in section III, where we use the Ward identity to construct a quasilocal operator that has support only at the positions of the electrons.

The vertex operators introduced above are primary fields, and as discussed by Moore and Read(2), the conformal block given by primary fields, such as (5), yields a ”representative” wave function for the QH state in question. In the simplest cases, these are the actual ground states for idealized (singular) Hamiltonians, but in general they are not known to correspond to any potential.
The importance of the representative wave functions lies instead in their topological properties and in the assumption that they are ”adiabatically” connected to the correct ground state. This raises the question of how to modify the representative wave functions without destroying their good topological properties. To answer this, Moore and Read first pointed out that to each primary field there is a ”conformal tower” of descendant fields, which have the same charge and the same braiding statistics. They then conjectured that the actual wave function, for a realistic potential admitting a QH liquid ground state, would lie in the space spanned by correlators of fields in such conformal towers, and thus has the same topological properties as the representative wave function obtained from the primary fields only. The descendants of a primary field are obtained from the OPE of this field with the energy-momentum tensor, which for the scalar fields considered so far is given by T(z)=12:(i∂φ(z))2:.
In general, descendants can be expressed as derivatives of the primary field V(z). In the present paper, we will only be concerned with the simplest case, namely first descendants; the first descendant of an arbitrary primary field Φ is simply a holomorphic derivative,

(L−1Φ)(z)=12πi∮zdwT(w)Φ(z)

=

∂zΦ(z).

(6)

One should, however, note that replacing a primary (electron) operator with a descendant will also change the angular momentum. Thus to construct a set of trial wave functions at a fixed L, we must restore the angular momentum e.g. by multiplying with Zcm=∑izi to the appropriate power. Another, more physical, approach is to create local particle-hole pairs using the operators H(η) and P(¯η). The precise relationship between these approaches remains to be clarified,
and one should also understand how they relate to the methods developed using composite fermions. 2

Descendant fields are not only important for constructing improved wave functions, but they are in general also needed to get a representative wave function in the first place.
In order to illustrate this point, and show how to go beyond the Laughlin states, we end this subsection with an example from the next level in the hierarchy. The simplest state at the second level is the ν=2/5 composite fermion state in the positive Jain series.
Describing the state requires two independent electron operators(4), which, in a suitable basis3, may be expressed as

V1(z)

=

ei√3φ1(z)

(7)

V2(z)

=

∂^V2=∂e2i√3φ1(z)+5i√15φ2(z).

Here, φ2 is an additional, independent bosonic field, described by the same action (3) as φ1.
In the following, primary fields are denoted by ^Vj, while the Vj’s denote electron operators and are in general descendent fields.
The ν=2/5 ground state is given by(4)

Ψ2/5=A⟨N/2∏i=1V1(zi)N/2∏i=1V2(zi)Obg⟩,

(8)

where A denotes antisymmetrization over the electron coordiates {zi}. In other words, the ground state is not just a single conformal block as in the Laughlin case, but rather an antisymmetrized sum over correlators involving two different representations of the electron.
In analogy to the composite fermion picture, these two representations can be thought of as corresponding to composite fermions in the first and second CF Landau level, respectively.
Since our formulation is entirely within the lowest Landau level, what
distinguishes the two operators V1 and V2 is the presence of the
derivative, which is naturally interpreted as giving the particle an
extra ”orbital” spin. This is in turn reflected in a shift in the
relation between flux and particle number when transcribing the wave
function eq. (8) to the sphere.

There are also two independent hole operators, determined for instance by the conditions(16),
⟨Vα(z)Hβ(η)⟩∼(z−η)δαβ,

H1(η)

=

e(i/√3)φ1(η)−(2i/√15)φ2(η)

(9)

H2(η)

=

ei(3/√15)φ2(η),

and quasihole states are obtained by inserting these in the correlator (8).
Less dense QH states at level 2 are obtained by modifying the coefficient of φ2 (in such a way that V2 remains fermionic). For example, choosing ^V2(z)=e2i√3φ1(z)+11i√33φ2(z), one obtains a candidate wave function for ν=4/11(5). In general, at level n of the hierarchy the ground state wave functions are constructed from n different electron operators, containing n bosonic fields, and with one additional derivative per level.
In this way, one can construct explicit wave functions for all hierarchical states corresponding to condensates of quasielectrons; explicit expressions for these wave functions are given in Ref. (5).

ii.2 The background charge

We now discuss a somewhat subtle, yet important, technical question, namely that of the background charge Obg. The need for such a background charge is most easily understood in the Coulomb-gas formalism where the vertex operators are fields that carry charge with respect to (in general several) U(1) groups, and the background is needed for the total system to be neutral, and thus to satisfy Gauss’ law. In open systems, such as a QH droplet with an edge, one can place a single large charge at a very large distance, Z, from the other particles and then extract the correlation functions by carefully taking the limit Z→∞(17).4
At a technical level, the disadvantage of this method is that it cannot be used in finite geometries, such as the torus, or for open geometries different from the circular disc.

A physically more appealing scheme, that also generalizes to other geometries, is due to Moore and Read(2) who proposed to use a continuous background charge distribution. For the simplest case of the Laughlin state, this is obtained by using

Obg=e−i√mρm∫d2xφ(z),

(10)

where ρm=ρ0/m, and ρ0=eB/2π=1/2πℓ2 is the density of a filled Landau level. The generalization to higher hierarchy levels involves several bosonic fields and is straightforward. This choice has the additional advantage of producing the correct Gaussian factor(2) (see below), so the correlators in fact give the full lowest Landau level wave functions.

Using the background charge (10), however, involves some technical problems. A naive calculation will result in integrals of the form ∫d2zln(z−zi), which are not properly defined. There are several ways out of this quandary. The simplest is to consider correlators of full, as opposed to holomorphic, vertex operators, and at the same time use the full field φ(z,¯z) in (10). Then the relevant integrals in the correlators are of the form ∫d2zln|z−zi|, which are well defined up to boundary terms. This approach has allowed us to construct hierarchical states on a torus, but the procedure for extracting the wave functions is less straightforward(18).

A second, purely holomorphic, approach was introduced in Ref. (4), where we replaced the continuous background charge with a regular lattice, z→n=(nx+iny)a, of K=kN charges, each of magnitude −m/k. The lattice spacing, a, is related to the magnetic length by a2/ℓ2=2πm/k, so the limit k→∞ corresponds to taking the lattice spacing to zero.
In Appendix A.2 we outline an extension of this method that allows for calculation of correlators involving insertions of the U(1) current and the energy momentum tensor.

We shall now introduce yet another way to treat the homogenous background charge which will prove to be very convenient in the following. The basic idea is to notice that the exponent of (10) is linear in φ, and thus can be included as part of the quadratic action by a field-independent shift. This method has the advantage of using only continuum fields, and will also provide a rigorous setting for the construction of the quasielectron operator in the next section. The following discussion will be entirely in the context of the ν=1/m Laughlin states, but the generalization to the hierarchical states is straightforward. The ideas we present here are presumably well known to experts in CFTc, but have to our knowledge not been used in the context of QH physics.5

Let us introduce the shifted field ~φ(z,¯z)≡φ(z,¯z)−iπρ¯zz
and rewrite the full bosonic action according to

where we neglected possible boundary terms. The currents and vertex operators are recast accordingly as

V(z,¯z)

=

ei√mφ(z,¯z)=ei√m~φ(z,¯z)e−√mρπ|z|2≡~V(z,¯z)e−√mρπ|z|2

(12)

J(z)

=

i√m∂φ(z,¯z)=i√m∂~φ(z,¯z)−πρ√m¯z≡Jp(z)+Jbg.

From the first line in (12), noting that πρ√m=1/2ℓ2, we see that the exponential factors needed for the QH wave functions are manifest. The second line in fact shows how the U(1) neutrality condition is implemented, in that the current is decomposed into a piece Jp due to the point-like particles, and a piece Jbg coming from the background charge. The total charge is obtained by integrating along a contour enclosing the QH droplet,

Q=1√m12π∮dz∂φ(z)=~Q−N

(13)

where we used ρ/√m=ρ0/m=N/A
with N the total number of electrons, and A the area of the droplet. Thus the original charge neutrality condition Q=0 now becomes

~Q=N.

(14)

This argument might seem too simple minded since we have been very cavalier about boundary terms, and one might also reasonably ask precisely how the charge neutrality is implemented in the new variables. In Appendix A.1 we present a Hamiltonian analysis, which does not make any assumptions about the boundary, and which explicitly demonstrates how the U(1) symmetry is implemented on the quantum states. There we also show that the shift (12) can be implemented at the level of holomorphic vertex operators by

V(z)

=

ei√mφ(z)=~V(z)e−|z|24ℓ2.

(15)

The relation (15) will be used in the next section. This way of incorporating the background charge in the action is completely consistent with the previously introduced method of regularizing the background charge by introducing a lattice of fluxtubes. The latter is elaborated in Appendix A.2, and the reader may convince herself that both approaches yield the same wave functions.

With the preliminaries in place, we are ready to address the central issue of this paper – how to construct an operator that directly creates a quasielectron localized at a specific point η.
In order to present the basic ideas of our construction in the simplest possible way, we will first discuss the Laughlin states. The generalization to the hierarchical states involves several technicalities and will be deferred to Section V.
In order to motivate our construction, we start by discussing the qualitative physical picture, before putting it into a formal language. First, recall the case of quasiholes.
As mentioned in section II, the one-quasihole wave function can be constructed by inserting a local quasihole operator, H(η)=e(i/√m)φ(η), into the correlator (5):

Ψ(L)1qh(η;z1…zN)

=

⟨H(η)V(z1)……V(zN)Obg⟩

(16)

=

N∏j=1(η−zj)ΨL(z1…zN)e−14mℓ2|η|2.

(17)

A naive guess for constructing trial wave functions for quasielectrons would be to insert the inverse operator, H−1(η), into the correlator. This operator has the correct charge and conformal dimension – but it does not yield appropriate fermionic wave functions, as the correlator includes singular factors, such as ∏i(η−zi)−1. The origin of the singularities is the Pauli principle, which makes it impossible to create the excess charge at an arbitrary point, as there is a finite probability that there is already an electron at this position.

However, there is a well-defined procedure to generate a localized excess charge, namely by shrinking the correlation hole around one of the electrons, thus allowing the electron liquid to become denser. Shrinking the correlation hole can be thought of as attaching an antivortex, and it creates the expected excess charge, em. As discussed in Ref.(4), this construction naturally leads to wave functions of quasielectrons in good angular momentum states. However, the excess charge can be located at an arbitrary position η by building linear combinations of attaching the antivortex at different electrons, weighted with a gaussian weight.
In doing this, we have to be careful not to introduce an unwanted antiholomorphic dependence on the electron coordinates w.
To proceed, we write
|w−η|2=|w|2−2w¯η+|η|2+(w¯η−¯wη), and note that since the term in the parenthesis is purely imaginary, it exponentiates to a pure phase.
Removing this phase, we can thus use the remaining terms to localize the excess charge in a region of the size of a magnetic length ℓ around the point w=η by introducing a factor e−14mℓ2(|w|2+|η|2−2¯ηw). The constant 1/4m was picked so that e−14mℓ2|η|2 is the exponential factor appropriate to a particle with charge q=−e/m moving in the magnetic field, and the remaining gaussian term e−14mℓ2|w|2 is precisely what is needed to eventually produce the correct gaussian factor for the electrons.

In order to find an operator that performs the desired attachment of the inverse hole, it is useful to recall the Ward identity (II.1). The divergence of the conserved charge current has only support on charged sources, in our case at the electron and quasihole positions, and it can therefore be used to place H−1 at the electron positions. However, to obtain well-defined expressions a regularization procedure is needed. In the simplest case, i.e. the Laughlin states, this regularization is nothing but the usual normal ordering. The hierarchical states at level n>1 require a generalized notion of normal ordering, which will be discussed in Section V. For the Laughlin states the quasielectron operator can be written in the following form

P(¯η)

=

∫d2we−14mℓ2(|w|2+|η|2−2¯ηw)(H−1¯∂Jp)n(w)

(18)

where Jp(w)=i√m∂~φ(w) is nonzero only at the electron positions, and does not see the background charge. We will use the notation P(¯η) rather than P(η,¯η) since the η dependence in (18) amounts to a trivial z-independent normalization. (The reason for choosing this convention will be clear in a later section.)
We can now use the Ward identity (II.1) to evaluate the correlators ⟨Jp(w)V(zi)⟩ to get delta functions, δ2(w−zi),
at the electron positions; the w-integral in turn gives terms ∼(H−1V)n(zi), with normal ordering symbol, (…)n, defined by

(H−1V)n(zi)≡∮zidzz−ziH−1(z)V(zi),

(19)

where the contour is close enough to zi not to enclose any other singularities(10).
For free fields, the normal ordering (19) coincides (up to an overall constant) with the normal ordering :⋯: used to define the vertex operators in (1).

That P(¯η) as defined above is an eigenstate of the total charge follows directly from the commutator, [Q,H−1(η)]=−1mH−1(η), while the quasilocal nature of the charge is revealed by studying the commutator with the localized charge operator

Q(η;ϵ)=1√m12π∮ηdz∂φ(z),

(20)

where the contour is a circle of radius ϵ around η. We get [Q(η;ϵ),P(¯η)]≈−1mP(¯η) up to an exponentially small correction for ϵ≫ℓ.

Since Jp is expressed in the shifted field ~φ, it is convenient to use the relations (12) to express the original vertex operators, V, in terms of the ~V’s, with the understanding that the expectation values are now taken using the action
S[~φ] (11), and with the modified charge neutrality condition (14). In this way we directly calculate the polynomial part, Ψ(hol)1qp(¯η;z1…zN), of the full wave function,

Ψ(hol)1qe(¯η;z1…zN)=⟨P(¯η)~Vm(z1)……~Vm(zN)⟩

(21)

with

Missing or unrecognized delimiter for \right

(22)

The formula (22) and its generalization (35) is one of the main results of this paper.

It was shown in Ref. (4), that the CF wave function for a single quasielectron localized at the origin, in the ν=1/m Laughlin state is given by

Ψ(CF)1qe(η=0;→r1…→rN)

=

A{e−|z1|2/4mℓ2⟨P(z1)V(z2)……V(zN)Obg⟩}

(23)

=

e−∑i|zi|2/4ℓ2∑i(−1)i∏j<k(i)(zj−zk)m∂i∏l≠i(zl−zi)m−1

=

e−∑i|zi|2/4ℓ2⟨P(¯η=0)~Vm(z1)……~Vm(zN)⟩

where the operator P(z) is defined as

P(z)=(H−1V)n(z)=∂ei(√m−1√m)φ(z).

(24)

Thus, inserting the nonlocal operator P(¯η) can be written as a sum over all electron positions zi of the local operator P(zi). Multiple insertions will naturally lead to multiple sums over (different) electron positions. A detailed calculation of the two-quasielectron wave function can be found in App. B.
In the earlier construction in Ref. (4), we were to neglect the non-holomorphic terms ∼¯zi coming from the derivative acting on the exponential part of the correlation function. This prescription was introduced ad hoc in order to get wave functions in the lowest Landau level. Using instead the quasielectron operator (22) located at the origin to evaluate (21), we reproduce the holomorphic part of (23) exactly.
In particular, the derivatives will act only on the polynomial part, so we automatically get holomorphic wave functions without invoking any ad hoc rule.
In the following we shall, unless stated otherwise, use the shifted fields (12), and for simplicity of notation we shall suppress the tildes and write V instead of ~V.

At this point we should stress a few important points.

The derivative occurring in (23) is a direct consequence of ”fusing” the operators H−1 and V using the standard normal ordering prescription defined by (19), which amounts to extracting the leading non-singular term in the OPE. There is however no a priori reason why we should use this prescription. The only formal requirement on the fusion is that the resulting local operator has the same U(1) charge and the same conformal dimension as the original composite operator. The first condition follows from charge conservation,
and the second from the composition properties of the (orbital) spin. The latter has a geometric meaning in that it gives rise to Berry phases on curved manifolds, and in particular determines the shift on the sphere which is a topological invariant(19).
In the hierarchical states where there are several U(1) symmetries, conservation of the charges is needed to get correct charge and (mutual) statistics of the quasiparticles.

From (23) we see that the final expression for the electronic wave function is in terms of correlators involving primary fields ^V of the rational CFT and descendants of the form Ln−1^V ; as we shall see below, the same will be true for the multi-quasielectron case. This structure is important for two reasons. First, it guarantees that the wave functions are analytic as they should be. The second point relates to the discussion following Eq.(6) – correlators containing descendants have the same braiding properties as those with the corresponding primary fields.

We do not expect that inserting P(¯η1)P(¯η2) will describe a state of two localized quasielectrons, since the operator P(z) in (24) is anyonic, as can be readily seen from the OPE P(zi)P(zj)∼(zi−zj)(m−1)2/m.
Below we shall explain how to supply the phase factors that are needed to obtain well defined electronic wave functions.

It turns out that these points are connected, and they are all relevant for finding a P operator that can be used at any level in the hierarchy. We will return to this, essentially technical, point later in the paper.
Finally, we comment that the scheme presented here is manifestly in
the disk geometry. It would obviously be of interest to carry out this
construction, including the background charge, in finite geometries. We
have so far not done this for the sphere; for some recent progress on
the torus, see Ref.(18).
In the following section we discuss a question of principal importance: How do we construct localized states of several quasielectrons, and how do we determine their statistics?

One of the most interesting aspects of quantum Hall quasiparticles is their fractional statistics. While the anyonic statistics of Laughlin quasiholes is very well understood, the situation is less clear for quasielectrons in the Laughlin state as well as for any quasiparticle in a general hierarchy state. This section is devoted to a discussion of these issues, and in particular the statistics of quasielectrons.
We begin with a brief overview of what is known, and what is believed, about the statistics of the quasiparticles in the Abelian hierarchy states. We then turn to a more detailed discussion of fractional statistics within the CFT approach, and
the specific problems we encounter when seeking to generalize the formalism of the previous section to many-quasielectron excitations, both of the Laughlin states and the more general hierarchical states.

We shall here briefly review what is theoretically known about the statistics of the quasiparticles in the Abelian hierarchical states, and discuss in turn holes in the Laughlin states, holes in hierarchical states, and finally quasielectron states.

Laughlin quasiholes

The anyonic nature of the Laughlin quasiholes is very well understood. The original calculation of the pertinent Berry phase by Arovas et al.(20) confirmed the assertions made earlier by Halperin(21) based on a specific ansatz for the hierarchical wave functions, and later it was realized that the fractional statistics could also be encoded in an effective topological low energy theory of the Chern Simons type(22).
The calculation of the Berry phase assumed that the QH liquid was incompressible, which in turn was proven by Laughlin using the plasma analogy(23). Alternatively, one can directly use the plasma analogy to compute the normalization factor N(η1,…,ηn) of a state with n widely separated quasiparticles, from which the exchange Berry phase can be extracted. In both cases, the arguments crucially depend on the plasma analogy, which is applicable only when the holomorphic part of the wave function is of Jastrow form. The fractional statistics of the Laughlin holes has also been verified by numerical simulations on small systems(24); (25).

Here we should make a technical comment that will be important in the subsequent discussion of the CFT approach: For the Berry phase to be uniquely defined, the wave function must be a single valued function of the adiabatic parameters. In the present context, this translates into the condition that the electronic wave functions must be single valued functions of the holomorphic hole coordinates. If the normalization constant of the electronic wave function for example contains a ”monodromy” factor (η1−η2)ϑ/π, the statistical phase pertinent to the exchange of the two quasiholes, will be the sum of ϑ coming from the monodromy, and the Berry phase.

In addition to the explicit calculations, there is also a very general gauge argument due to Kivelson and Rocek, that ties the fractional statistics of a Laughlin hole to its fractional charge(26). It goes as follows: At the center of a Laughlin hole the electron density is zero, and thus we can imagine that it supports a thin tube of unit flux – in fact this was how Laughlin originally argued for the existence of a fractionally charged quasihole. Now imagine transporting another quasihole around the one with a unit flux at the center. Since the quasihole has charge q=e/m, it will pick up an Aharonov-Bohm phase equal 2π/m. But the system is made up only of charge −e electrons all moving entirely in the flux free region, so we can appeal to the Byers-Yang theorem (27) to conclude that the wave function has to be single valued if the flux equals an integer number of elementary flux quanta. Consequently there must be another phase to cancel the fractional part of the Aharonov-Bohm pase, and this is precisely the fractional statistics phase θ which is thus equal to π/m (the factor of 2 is because a path corresponding to one particle encircling the other is equivalent to two exchanges). This means that the essentially non-local phenomenon of fractional statistics is closely related to the local phenomenon of fractional charge. The latter is both easier to understand theoretically and simpler to measure in the laboratory(28).

Quasiholes in hierarchical states

Unfortunately the Laughlin states are among the very few where the Berry phases related to quasiparticle exchange can be calculated analytically without additional assumptions. The other important class are the Halperin (n1,n2,m) states(12) that describe multilayer systems. In all cases the wave function is a single product of Jastrow factors, which means that the powerful plasma analogy may be applied.
For states that are not of this type, the arguments for fractional statistics are much weaker. In the original hierarchy proposal by Halperin, the claim was based on the analysis of the hierarchical wave functions, which however are too complicted to allow for an explicit calculation of Berry phases. In a later paper Read analyzed the statistics of the hierarchical states in a more precise way, but had to make a ”orthogonality postulate”, that was made plausible but was not proven(35).
Another line of arguments is based on effective Chern-Simons theories, but unlike the Laughlin case, these cannot be derived from the microscopic theory. The third line of argument, which we shall scrutinize more closely below, is based on CFT.

What, if any, are the implications of the Kivelson-Rocek (KR) argument in this case? First we should note that the argument is certainly not always applicable. Take for instance the charge q=e/4 quasiholes in the Pfaffian state, which do not obey the expected π/4 Abelian statistics, but are known to have non-Abelian statistics.
The difficulty lies in the fact that the electron density does not vanish at the position of the hole, as can be seen from the corresponding OPE.
The same holds for the q=e/5 quasiholes in the 2/5 state, so we cannot directly use the KR argument. There is, however, another line of argumentation that strongly suggests that, in the absence of degeneracies, the elementary quasiholes and quasielectrons do obey pπ/5 Abelian fractional statistics for some integer p. First we note that according to Laughlin’s original gauge argument one can always form a hole with charge q=νe by inserting a thin flux line. For such holes the KR argument holds and we can conclude that the statistics is Abelian with a phase νπ. Alternatively we note that the Berry phase calculation for the Laughlin quasihole of Arovas et al. is valid for any incompressible state(20).
Next assume that the Laughlin hole can split into two or more fundamental holes, in this case two charge q=e/5 elementary holes. Then we can use a general argument, due to Thouless and Wu, which on the basis of locality suggests that the statistical phase accumulated when a particle encircles a cluster of particles is just the sum of the phases corresponding to encircling the constituents(29). In CFT language this is reflected in the Abelian fusion rules of the underlying U(1) current algebras. The clustering conditions do not uniquely define the statistical angles, but do give the constraints θ11=θ22 and
θ12+θ21=π/5. Using the hole operators in Eq. (9), and noting that the combination H1H2 corresponds to a Laughlin hole, we find θ11=θ22=3π/5 and θ12=θ21−2π/5, which is consistent with the above conditions.
6
If we would further demand that also H1H1 and H2H2 should have the same statistics as a Laughlin hole, the solution follows uniquely from the clustering conditions. In an early paper, Su used this stronger condition to get θ=3π/5, but without considering the possibility of two non-equivalent holes(30).
Although we used the 2/5 state as an example, we believe that the above types of arguments would also constrain the possible statistical angles of the fundamental quasiparticles in a general hierarchical state.

Quasielectron states

The status of the quasielectrons is less clear, even for the Laughlin states. First there are two rather different proposals for the Laughlin quasiparticle.
The first is Laughlins original proposal,

Ψ(L)1qe(¯η;→r1…→rN)=N∏i=1(∂i−¯η)ΨL(zi…zN)

(25)

which is closely related to the corresponding Laughlin quasihole(23). The other is the composite fermion wave function (23). In neither case is there any simple argument even for the fractional charge. (The gauge argument which works beautifully for the quasihole is not applicable since it yields a wave function that has components in the second Landau level, and has to be projected.) There is however convincing numerical evidence for the charge to equal −e/m. The situation concerning the fractional statistics is less clear. Extensive calculations by Kjønsberg and Leinaas(25), and by Jeon, Graham, and Jain(31), have shown that the two-quasielectron wave functions of the Laughlin and Jain type differ substantially in that the latter exhibits a clearly defined statistical angle for well separated quasielectrons, while the former does not. It can of course still be true that the Laughlin quasielectrons have the expected statistics for much larger separation, but that remains to be shown.

Turning to the hierarchical states, the general arguments based on Chern-Simons theories or the explicit form of the hierarchical wave functions, can be made also for quasielectrons, and are subject to the same type of criticism. It should be clear that the Kivelson-Rocek argument is not applicable in this case, but by using the Thouless-Wu argument and comparing a quasihole taken around a quasielectron with that of a quasihole taken around another quasihole, we can conclude that the mutual fractional statistics between quasielectrons and quasiholes only differs from that of quasiholes by a sign. Then by noting that having a quasielectron go around a quasielectron - quasihole pair is a trivial operation, we conclude that the statistics of a quasielectron equals that of a quasihole with the same charge. So although one should keep in mind that there is no plasma analogy, and thus no conclusive analytical calculation, the arguments for the fractional statistics of the quasielectrons are nevertheless quite convincing,

iv.2 Quasiparticle statistics in the CFT approach

We now turn to the CFT approach to QH wave functions, and again start, in the first two paragraphs, by discussing the quasihole states, before addressing quasielectrons in the remainder of this section.

Laughlin quasiholes

Before addressing the question of two Laughlin quasielectrons, we remind ourselves of the corresponding case of two quasiholes. The pertinent correlators are of the form

⟨H(η1)H(η2)V(z1)…V(zN)⟩∼(η1−η2)1mF2h(η1,η2;z1…zN),

(26)

where V(z) is given by Eq.(1), F2h is an analytic function of all its arguments, and θ=1/m is the fractional statistics parameter of the quasiholes created by H. Referring back to section IV.1.1, we note that this wave function is not single-valued in the quasihole coordinates, so the statistical phase is no longer given by the Berry phase only. In this case the full statistical phase is given by the monodromy of the correlation function, or the conformal block. This in turn implies that the Berry phase has to be zero, which can easily be explicitly verified using the plasma analogy.

Quasiholes in hierarchical states

As a concrete example, let us again look at the ν=2/5 state which according to (8) can be expressed as a correlator of two types of electron operators. Quasihole excitations of this state are obtained by inserting a number of the charge q=e/5 hole operators H1 and H2 given by (9). In parallell with the Laughlin case, there are monodromies (η1−η2)θαβ, with θ11=θ22=3/5 and mutual statistics θ12=θ21=−2/5, which is consistent with the expected fractional statistics of the quasiholes(15). Here we should comment on that our trial wave function eq.8 is not a single correlator of the form eq.26, but an antisymmetrized sum of such correlators.
Since, however, the holes have trivial braidings with the electrons, the monodromies will be the same for all terms in the sum,
and are thus well defined for the full wave function eq. (8).
The same will be true in a general hierarchical state.
This, however, does not constitute a proof of the fractional statistics, since we cannot compute the Berry phase, but simply have to assume that it is zero, just as in the Laughlin case. This assumption of having zero Berry phases related to quasiparticle exchange when the wavefunctions are written as CFT correlators, was first tacitly made in the seminal paper by Moore and Read(2). A heuristic argument for why this should be true was given in Ref. (32), and an attempt to construct a generalized plasma analogy using the Coulumb gas formulation of CFT was discussed in Ref. (33). Unfortunately, none of these arguments can, in our view, be considered as proofs of a zero exchange Berry phase.
Recently Read has given an analytic argument for the vanishing of the Berry phases for a class of non-Abelian wave functions that can be expressed as a single conformal block(34). Although quite general in nature, it is not clear to us whether this kind of argument can be applied also to the states at higher levels in the Abelian hierarchy, where the wave functions are sums of conformal blocks.

It is certainly not true that the Berry phases corresponding to any choice of the quasihole operators vanish. To understand this, recall that
within the CFT framework the fractional statistics is reflected in the OPE of the hole operators, which is directly related to the monodromies of their conformal blocks.
For example, for the Laughlin states we have H(η1)H(η2)∼(η1−η2)1/m, and precisely this factor also enters the full correlation functions involving these operators, giving the monodromy and thus the statistics.
There is however a large freedom in choosing the quasihole operators without changing their long-distance physical properties such as charge and statistics. Taking the ν=1/3 Laughlin state as an example we can, instead of taking H(η)=eiφ(η)/√3, use one of the following operators,

Hb(η)

=

eiφ(η)/√3−i√5/3χ(η)

(27)

Hf(η)

=

eiφ(η)/√3−i√2/3χ(η),

where χ is a new free scalar field with the same normalization as φ. In the two-quasiparticle wave function, this will correspond to the replacements (η1−η2)1/3→(η1−η2)2 and (η1−η2)1/3→(η1−η2) respectively, which shows that Hb is a bosonic operator while Hf is fermionic. Since this only corresponds to an overall re-phasing of the electron wave functions, the physics of the quasiholes is clearly unchanged and we have just shuffled the statistical phase from the monodromy to the Berry phase. This freedom in choosing the monodromies of the quasiholes is well known, and was, in a different language, pointed out in an early paper by Halperin(12).
Also, everything said so far directly generalizes to holes in arbitrary hierarchical states.

We can now make the conjecture about vanishing Berry phases more precise: Quasihole wave functions that are written in terms of conformal blocks of a collection of hole operators H and N electron operators V, have vanishing Berry phases corresponding to the braiding of the quasiholes for large N, as long as all fields in the quasihole vertex operators are also present in the electron operator. In the hierarchy case where there are several electron operators Vn and different hole operators Ha, all fields in the Ha’s must occur in at least one of the Vn’s, and the number of electrons, Nn in all the groups must be large.

At this point many readers might think that we are stressing some rather obvious points, but we shall see that in the case of quasielectrons things are more complicated than for the quasiholes. Within our CFT scheme, the choice of hole operators will no longer be arbitrary, and we can in general not expect to read the statistics from the monodromy. This will be especially important in the subsequent discussion of the non-Abelian Moore-Read state.

Two quasielectrons and the quasielectron - quasihole pair

All the basic problems related to generalizing the approach in section III to many-quasielectron states occur already for the simplest case of two quasielectrons or one quasielectron - quasihole pair in the Laughlin state. The general underlying problem was pointed out already at the end of section III — when we use (22) to fuse an anyonic inverse hole operator with an electron, the resulting operator P(z) is also anyonic and will introduce an unacceptable non-analytic behavior in the electron coordinates. To take a specific example, the quasielectron counterpart to (26), using the operator P (22), becomes

⟨P(¯η1)P(¯η2)V(z1)…V(zN)⟩∼∑i<j(−1)i+j∂i∂j(zi−zj)1mF2p(η1,η2;z1…zN).

(28)

As expected, this wave function is not analytic in the electron coordinates zi and does thus not describe a state in the lowest Landau level.

At a qualitative level this problem is quite easy to deal with. Because of the localization implied by Eq.(18), the sum is dominated by terms where zi and zj are within a magnetic length away from η1 and η2 respectively. For widely separated quasiparticles, i.e. |η1−η2|≫ℓ, we can thus make the replacement (zi−zj)→(η1−η2), to get a function which is analytic in the zi’s. Note that the analytic function F2p contains factors ∏k≠i,j(zi−zk) which will not be modified, and it is these factors that determine the fractional charge.
Although most likely correct for widely separated quasielectrons, this treatment is unsatisfactory for two reasons. First it is hard to estimate the errors involved when η1 and η2 approach each other. In particular this means that such wave functions will not easily accommodate the dense quasiparticle condensates that are the building blocks of the hierarchy, as explained in the next section. Secondly, since the space of states spanned by the correlators of the operators V, H, and P is not holomorphic, it means that the operator P cannot easily be thought of as a quasilocal counterpart to the local holomorphic vertex operators V and H, and this removes much of the formal appeal of our approach.

The difficulties related to the factor (zi−zj)1m on the right hand side of (28) were disucssed already in Ref. (4), and two alternative approaches were proposed. The first was to introduce by hand a suitable factor (zi−zj)n−θπ in the wave function. This is not appropriate here since it would amount to replacing the product P(¯η1)P(¯η2) by a new, essentially non-local, operator that directly creates two quasielectrons. The other approach was to change the inverse hole operator H−1 to become bosonic, or alternatively fermionic, so that the resulting P operator becomes fermionic or bosonic, respectively.

It should now be clear how to proceed; the latter approach simply amounts to using the operators Hb and Hf in (27) respectively, in the expression for P. One should however note that, as opposed to the case of quasiholes,
this modification of the operator does not result in a mere re-phasing of the electronic wave functions. For instance, using Hb, the factor (zi−zj)13 in (28) is replaced by the holomorphic factor (zi−zj)2 which amounts to changing both the phase and modulus of the wave function. In Appendix B we present an explicit calculation of the two-quasielectron wave function in this basis, along with further technical details.

Using a bosonic or fermionic form of the inverse hole operator in P, the resulting wave functions will be holomorphic in the quasielectron coordinates, so the fractional statistics will reside entirely in the Berry phase.
Remember that in the case of quasiholes, the fractional statistics could be read off directly from the monodromies of the anyonic operators, while introducing the auxiliary fields yielded wave functions that were holomorphic in the quasihole coordinates with the statistics ”hidden” in the Berry phase. We now propose that the same is true for the quasielectrons, i.e. that their fractional statistics can be obtained directly from the monodromies of the anyonic operators, H−1, which do not contain the extra bosonic field χ.
This amounts to a rather natural extension of the corresponding assumption about vanishing Berry phases in the case of quasiholes. A note of caution is needed here. Since changing the monodromies of the quasielectrons is not merely a rephasing of the wave function, but changes the correlations between the electrons, using different representations leads to slightly different charge profiles. This is only a short distance effect; large distance properties, such as charge and statistics, should be insensitive to this.
We should also mention that in Ref. (4) we presented a calculation of the Berry phase of two Laughlin quasielectrons using a random phase approximation, but it is not clear how to extend this to higher levels in the hierarchy.

Note that changing the statistics of the inverse quasihole is done with help of an uncharged (in terms of electric charge) bosonic field, and in particular there is no homogeneous background charge that has a screening effect on this additional field.
In order to properly define the correlators, we must introduce a compensating charge also for the field χ.
Formally we can do this by again putting a continuous background charge, but this is quite unnatural from the physics point of view, and it is also formally unsatisfactory in that it gives the wrong exponential factors in the wave functions.
Alternatively, we can neutralize the correlator by putting a compensating charge at a large distance and take the appropriate limit as explained for instance in Ref. (17). Neither approach is fully satisfactory, although the latter one complies more with the physical picture of creating a quasielectron together with a compensating quasihole or a charged edge excitation. In particular, if we want to view the background charge to be fixed by the ground state, then it should be kept unchanged for the few-quasiparticle case.7

The most natural way to achieve charge neutrality is to directly consider quasielectron - quasihole pairs. Using the original, anyonic, hole operators we have,

⟨P(¯η1)H(η2)V(z1)…V(zN)⟩∼∑i(−1)i∂i(zi−η2)−θπFph(¯η1,η2;z1…zN).

(29)

which is again not analytic in the electron coordinates, due to the prefactor (Fph is analytic in {zi}). This problem can be resolved using the same ideas as in the above case of two quasielectrons. Either we can by hand make the approximation (zi−η2)−θπ≈(η1−η2)−θπ, or we modify the hole operator according to

H=ei1√3φ1→Hf=ei1√3φ1−i2√6χ,

(30)

and at the same time use the fermionic inverse hole in the quasielectron operator. In this case we create a quasihole-quasielectron pair which does not violate the charge neutrality condition of the correlator. In order to neutralize the correlator, we could equally well have used a bosonic form of the quasihole. This would however not give an acceptable wave function, since the OPE Hb(η)V2(z)∼1/(η−z) has a singular term. In the case of quasielectrons only, or quasiholes only, there is however no problem with using the bosonic representation.

In summary, we believe that the CFT approach to QH wave functions has given strong support to the assignment of fractional statistics that was made earlier on the basis of effective CS theories, reasoning based on hierarchical ansätze, and on general topological arguments. We also suggest that our CFT approach to quasielectrons, based on quasi-local operators, provide arguments for the fractional statistics of the quasielectrons which are at the same level of rigor as those for the elementary quasiholes in the hierarchy states.

As explained in section II, states at level n in the hierarchy are constructed using n electron operators and involve correlators of n scalar fields. Since there are also n inequivalent hole operators Hi, there are n2 different candidates for the quasielectron operator (18), but, as we shall see, only n of these are non-vanishing. Furthermore, in the important case of the Jain states, only one of these n operators gives a non-zero result when inserted into antisymmetrized correlators of electron fields. A heuristic argument for this can be given in the language of composite fermions(4): in hierarchical states that are formed from a number of filled ”effective Landau levels” of composite fermions, the only way to form a quasielectron excitation is to put a composite fermion in an unfilled level. In the language of CFT, this amounts to using Hn, which creates a hole in the highest effective Landau level (defined by the electron
operator with the largest number of derivatives, or highest orbital spin), in the construction of P.

Even after having specifed which Hi to use in the construction, we cannot directly use the formula (18) to obtain P. As we shall see, the difficulty has to do with the use of the normal ordered product defined in (19), and the resolution lies in substituting this with a ”generalized normal ordering procedure”. This section is technical in nature and can be omitted by any reader who is willing to accept that one can consistently define a product with the property (38). We now proceed to prove this, starting from the simplest case of the ν=2/5 Jain state.

v.1 The ν=2/5 state

The ν=2/5 Jain state is at the second level in the hierarchy, and we refer back to section II for the definition of the operators V1 and V2=∂^V2 (where V1 and ^V2 are primary fields), and the hole operators, H1 and H2.

Following the logic of the previous section, we now form a bosonic version of the operator H2 by H2b=H2e−(7i/√35)φ3, and then insert one or more of the corresponding P(¯η) operators in a correlator containing an equal number of V1 and V2 operators. Before carrying out the w-integrals from the P operators, we
notice that the derivatives ∂i in the V2(zi) operators can all be taken out front, to leave us with a correlator involving the primary fields H2b, V1 and ^V2. The terms surviving the normal ordering (19) will involve the following structures

(H−12b^V2)(z)

=

(∂e−3i√15φ2(z)+7i√35φ3(z))e2i√3φ1(z)+5i√15φ2(z)

(31)

(H−12b^V1)(z)

=

e3i√3φ1(z)−3i√15φ2(z)+7i√35φ3(z),

(32)

as is shown by identifying terms ∼(z−w)0 in the OPE (4).
Note that the first line cannot be written as a total derivative of a vertex operator, and is thus not the first descendant of any primary field in the theory. This is a technical observation, but it has consequences for the resulting wave functions. Recalling the discussion following Eq.(6), we know that there are general arguments why it is desirable to construct wave functions from primary fields and their descendants only. Moreover, recall that by introducing the bosonic representation for the quasielectrons, all operators in our theory have conformal dimensions such as to ensure that the correlation functions of primary fields are analytic. This in turn implies that correlators of descendants of the form Ln−1Φ=∂nΦ where Φ is a primary field, are also analytic. The way this works is that correlators corresponding to the individual bosonic fields are not analytic but give cut singularities of the form (zi−zj)a, with a non-integer; these however combine to integer powers in the final product, e.g.9/3+9/15+49/35=5 in (32). Thus terms of the type (31), with the derivatives acting on one of the non-analytic factors only, will potentially generate pole singularities in the wave function.

A second difficulty is that we seemingly have two different types of quasielectrons, corresponding to the two operators (31) and (32). This might not seem like a problem, but it is in conflict with well-established results on the ν=2/5 state. In particular, this state is known to be very well described by the composite fermion approach, which predicts only one type of quasielectron; thus one would have hoped to reproduce those results. As shown in Ref. (4) , this is achieved if only the product (H−12b^V2) survives.

It turns out that both these problems can be resolved by a redefinition of the normal ordered product (19). This will at the same time put (H−12b^V1) equal to zero and convert the first line in (31) into a total derivative of a primary field. We can then show that the same will happen at all levels in the hierarchy: only one normal ordered product survives, and the resulting operator is the first descendant (or holomorphic derivative) of a primary field.

v.2 The generalized normal ordering

Given the above qualitative arguments against using the naive normal ordering procedure, let us go on to present a generalized normal ordering which does not suffer from these problems. To this end, consider again the OPE,

H−12b(w)^V2(z)

∼

1(w−z)e2i√3φ1+2i√15φ2+7i√35φ3

(33)

+

(−3√15i∂φ2(w)+7√35i∂φ3(w))e2i√3φ1+2i√15φ2+7i√35φ3+O(w−z).

As noted in the previous subsection, the standard normal ordering amounts to extracting the finite second term by the contour integral (19). Alternatively, we could choose to extract the first term, which is a primary field. This has the same U(1) charge as the composite operator on the left, but not the same conformal dimension.
If we demand that the surviving term of H−12b^V2 should have the same conformal dimension as the composite operator, we must supplement it with an operator of dimension 1. The only dimensionful parameters at hand are the quasielectron position ¯η and the magnetic length ℓ. Thus the only two possibilities are ¯η/ℓ2 and ∂i, since 1/ℓ would amount to a non-analytic dependence on the external magnetic field B∼ℓ−2. The term proportional to ¯η is allowed, but vanishes for a quasielectron at the origin, and can in general be shown to correspond to an edge term as discussed in Ref. (4). We are thus left with the derivative operator. We cannot, however, just replace H−1(z) by ∂zH−1(z). Instead we recall that the operator L−1 (6) has conformal dimension one and amounts to a derivative when acting on a primary field Φ(z). These considerations thus lead us to define a generalized normal ordering by (…)gn by

where the double contour integral is defined by the y-contour enclosing the w-contour. Note that in the Laughlin case, this is equivalent to the conventional normal ordering used in the previous section.

which should be compared to equation (22). It is now easy to verify that

(H−12b^V2)gn(z)

=

∂e2i√3φ1+2i√15φ2+7i√35φ3

(36)

(H−12b^V1)gn(z)

=

0.

Eq. (36) is exactly the operator which in Ref. (4) was found to reproduce the composite fermion wave functions. Thus, inserting one or more of the operators (35) into the the correlator (8) yields well behaved quasielectron wave functions of the composite fermion type.

v.3 General hierarchical states

Let us end this section by outlining how our construction generalizes to an arbitrary level n of the hierarchy. At the nth level one has n electron operators Vα, α=1,...,n, and corresponding hole operators Hα. The latter can always be chosen so that the only singular term in the OPE is given by

H−1α(z)^Vβ(w)∼1(z−w)δαβ^Vα+1(w),

(37)

where ^Vα+1 is essentially (i.e. up to the additional derivative coming from the normal ordering) the new electron operator that will occur at the next level in the hierarchy.8
When computing correlators containing P(¯η) and Vα’s, a pole from (37) is necessary for the y-integral in (35) to be non-zero; thus one gets a contribution only for α=β,